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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Intersection.OpenChartDegree
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.ProjectiveBertini MaximalSeshadri.Frames
variable {X Y : Scheme.{0}}
theorem pullback_section_zero_iff_local_coefficient
(p : X ⟶ Spec (CommRingCat.of ℂ))
(I : X.IdealSheafData) [IsIntegral I.subscheme]
(L : LineBundle X) (s : O X ⟶ L.sheaf)
(U : X.affineOpens) [Nonempty (I.subschemeι ⁻¹ᵁ U.1).toScheme]
(e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme) :
pullbackSection I.subschemeι s = 0 ↔
U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)) ∈ I.ideal U := by
let : Algebra ℂ Γ(X,U.1) := (openScalars p U.1).toAlgebra
let : Algebra ℂ Γ((I.subschemeι ⁻¹ᵁ U.1).toScheme,⊤) :=
(baseScalars ((I.subschemeι ⁻¹ᵁ U.1).ι ≫ (I.subschemeι ≫ p))).toAlgebra
obtain ⟨eF,he⟩ := exists_restricted_pullback_frame I.subschemeι U.1 e s
let π := structuralOpenChartMap p I.subschemeι U.1
let f := U.1.topIso.hom (coefficient e (restrictSection U.1.ι s))
have hcoeff : π f = coefficient eF
(restrictSection (I.subschemeι ⁻¹ᵁ U.1).ι (pullbackSection I.subschemeι s)) := by
rw [he]
change (I.subschemeι ∣_ U.1).appTop
(U.1.topIso.inv (U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)))) = _
rw [U.1.topIso.hom_inv_id_apply]
have hker : f ∈ I.ideal U ↔ π f = 0 := by
rw [← structuralOpenChartMap_ker p I U]
rfl
rw [show U.1.topIso.hom (coefficient e (restrictSection U.1.ι s)) = f from rfl,
hker, hcoeff]
constructor
· intro h
rw [h,restrictSection_zero,coefficient_zero]
· intro h
apply (L.pullback I.subschemeι).restricted_coefficient_injective
(I.subschemeι ⁻¹ᵁ U.1).ι eF
change coefficient eF
(restrictSection (I.subschemeι ⁻¹ᵁ U.1).ι (pullbackSection I.subschemeι s)) =
coefficient eF (restrictSection (I.subschemeι ⁻¹ᵁ U.1).ι
(0 : O I.subscheme ⟶ (Scheme.Modules.pullback I.subschemeι).obj L.sheaf))
rw [restrictSection_zero,coefficient_zero]
exact h
end
end MaximalSeshadri.Geometry
end
end
end
end OAI
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